The Fricke-Macbeath curve, also called the Macbeath curve, is a smooth projective algebraic curve of genus
7. Its complex points form a compact Riemann
surface, also called the Macbeath surface or Fricke-Macbeath surface. The conformal maps from the surface to itself form
an automorphism group isomorphic
to the projective special linear group 
, of group
order 
.
The curve is unique up to isomorphism among curves
of genus 7 attaining the Hurwitz
bound (Macbeath 1965, Top and Verschoor 2018).
An affine plane model over Q, attributed to Bradley Brock by Top and Verschoor (2018, p. 119), is
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The plane curve has 14 ordinary double points, and its projective closure has no singularities at infinity. Its normalization is the smooth Fricke-Macbeath curve (Top and Verschoor 2018).
The curve carries the Hurwitz map 
. The 1-skeleton of
this map is the Fricke-Macbeath graph (Bokowski
and Cuntz 2018).
Fricke introduced the associated Riemann surface in 1899, and Macbeath presented explicit algebraic equations in 1965 (Fricke 1899, Macbeath 1965, Top and Verschoor 2018).
of Genus 7: A Polyhedral Realization."
Art Discrete Appl. Math. 1, #P1.02, 2018.